M-Point Boundary Value Problem for Caputo Fractional Differential Eqautions

Transcription

1 OSR Joral o Egieerig OSRJEN SSN e: 5- SSN p: Vol. 6 e ebrar. 6 V PP Poi Bodar Vale Proble or Capo raioal Diereial Eaio J.A.Naware Depare o aheai Shririha ahaidalaa Gjoi- 65 Di.Oaabad.SNDA Abra: - Siie odiio or he exiee o olio or a la o -poi bodar ale proble iolig Capo raioal deriaie are eablihed ig ixed poi heore. Baah ixed poi heore Shaer ixed poi heo- re ad Lera-Shader pe oliear alerae are applied o d exiee rel. Kewor: - Bodar ale proble exiee rel Capo raioal deriaie ixed poi heore. NRODUCON Reel a reearher are araed owar raioal diereial eaio a a pheoea i ario brahe o iee ad egieerig are odeled. Ner- o appliaio are od i orol e io-elaii eleroheir phar- aoiei ood iee e [ ]. Sigiia oribio b reearher ha bee reorded i he oograph de o Kilba e al [6]. Soe rel o he heor o raioal diereial eaio de o Lahiaha e. al. a be ee i [7 8 9 ]. Periodi bodar ale proble iegral bodar ale proble ad iiial ale proble or raioal diereial eaio o order < < wa died repeiel b Rairez ad Vaala [] Wag ad Xie [] ad Zhag []. Ahor deeloped oooe ehod or e o raioal diereial ea- io wih ario pe o odiio iolig Riea-Lioille raioal deriaie ad Capo raioal deriaie o order < < ad obaied exiee ad iee rel. [ 5 8 9]. Behora [] i he ear 9 ob- aied iie odiio or he bodar ale proble. Reel ahor obaied iie odiio or he exiee o olio o bodar ale proble ig ixed poi heore [6 7]. hi paper iie odiio or he exiee o olio o he ollowig -poi bodar ale proble BVP iolig Capo raioal deriaie are eablihed ia ixed poi heore. D = o J = [ ] wih he bodar odiio = = = =... = where D i he Capo raioal deriaie : J R R i a oio io ad are real oa.. PRELNARES Noaio deiiio ad preliiar rel reired i he laer eio are died here. C J R deoe Baah pae o all oio io ro J io R wih he or := p{ : J}. Deiiio. [ 6] he raioal iegral o a io o order i deoed b. i deied a where Γ. i he Eler-Gaa io ad L [a b] R. Deiiio. [ 6] he Capo raioal deriaie o o order i deoed b D. i deied a eraioal orgaizaio o Sieii Reearh P a g e

2 -Poi Bodar Vale Proble or Capo raioal Diereial Eaio eraioal orgaizaio o Sieii Reearh P a g e D Z Deiiio. A io C J R wih i -deriaie exiig o J i aid o be a olio o he proble i aiie he eaio D o ] [ J ad he iiial odiio iii " ' o ollowig Lea pla ipora role i he exiee o olio or he BVP -. Lea. [] Le > he he raioal diereial eaio D ha olio i i i or oe i R i = = [] +. Lea. [] Le > he h D. or oe i R i = = [] +.. EXSENCE RESULS Exiee rel o he BVP - whih i a iediae oeee o Lea. ad Lea.. Lea. Le - < ad le : J R be oio. A io i a olio o he raioal iegral eaio!!! i ad ol i i a olio o he raioal BVP J h D = = = =... = 5 Proo: Ae ha aiie. Applig Lea. we obai h h ' h. 6 ' h.. 6 "'... h!

3 -Poi Bodar Vale Proble or Capo raioal Diereial Eaio eraioal orgaizaio o Sieii Reearh P a g e Uig iiial odiio we ge!.... h!! Hee. h h!!! Coerel ae ha aiie raioal iegral eaio he b deiiio o Capo deriaie i ollow ha eaio ad eaio 5 alo hol.. AN RESULS hi eio we obai rel baed o Baah ixed poi heore ad Shaer ixed poi heore. ollowig rel i obaied b ig Baah ixed poi heore. heore. Ae ha here exi a oa > h ha or eah J ad all ȳ R.! 6 he BVP.-. ha a ie olio o J. Proo: raor he proble - io a ixed poi proble. Deie he operaor : C J R C J R b!!!! Clearl he ixed poi o he operaor are olio o he proble -. We hall e he Baah oraio priiple o proe ha ha a ixed poi. Now we hall how ha i a oraio appig. Le C J R. he or eah J we hae +! 5!! h

4 -Poi Bodar Vale Proble or Capo raioal Diereial Eaio eraioal orgaizaio o Sieii Reearh P a g e! Coeel b eaio 6 i a oraio. B Baah ixed poi heore we lai ha ha a ixed poi whih i a olio o he bodar ale proble -. ollowig rel i baed o Shaeer ixed poi heore: heore. Ae ha i : J R R i oio ii here exi a oa > h ha or eah J ad all R. he he BVP - ha a lea oe olio o J. Proo: We hall e Shaer ixed poi heore o proe ha ha a ixed poi. Now we proe: a i oio: Le be a eee h ha i C J R. he or eah J +! Sie i oio io we hae a. b ap he boded e io he boded e i C J R: i eogh o how ha or a η > here exi poiie oa l h ha or eah Bη = { C J R : η} we hae l. B apio ii we hae or eah J!!!!!!!! l!! h l :!!! ap boded e io he eioio e C J R: Le J < B η be boded e o C J R a i b ad le B η. he +!!!

5 -Poi Bodar Vale Proble or Capo raioal Diereial Eaio eraioal orgaizaio o Sieii Reearh 5 P a g e +!!! +!!! A he righ had ide o he aboe ieali e o zero. Uig Arzela-Aoli heroe we olde ha : C J R C J R i opleel oio. d A priori bo: Now we how ha : oroe R J C E i boded. Le E he = λ or oe < λ <. h or eah J we hae! +!!! hi iplie b apio ii ha or eah J we hae!!! h or eer J we hae R. :!!! hi how ha E i boded. A a oeee o Shaeer ixed poi heore we olde ha ha a ixed poi whih i a olio o he bodar ale proble -. ollowig exiee rel or he BVP - i obaied b ig Lera-Shader pe oliear alerae. heore. Ae ha i : J R R i oio ii here exi ϕ L J R + ad oio ad odereaig ψ : [ h ha ϕ ψ or eah J ad all R. iii here exi a oa > h ha P V L 7

6 -Poi Bodar Vale Proble or Capo raioal Diereial Eaio where P= he he BVP - ha a lea oe olio o J. Proo: Deie he operaor a i heore. ad.. a be how ha i oio ad opleel oio. or λ [ ] le be h ha or eah J +! h L!! REERENCES [] R. P. Agarwal. Behora S. Haai Bodar Vale Proble or raioal Diereial Eaio Georgia aheaial Joral []. Behora S. Haai S. K. Noa Bodar Vale Proble or Diereial Eaio wih raioal Order ad Noloal Codiio Nol. Aal []. Capo Liear odel o Diipiio whoe Q i Alo depede Geoph. J. Ro. Aroo [] D. B. Dhaigde J. A. Naware V. R. Nia oooe ehie or Se o Capo raioal Diereial Eaio wih Periodi Bodar Codiio D. Coi. Di. pl. S. A 9 5a [5] A. Graa ad J. Dgdji ixed Poi heor Spriger oograph i aheai Spriger-Verlag New Yor. [6] A. A. Kilba H.. Sriaaa J. J. rjillo heor ad Appliaio o raioal Diereial Eaio Norh Hollad aheaial Sdie Vol.. EleierNorh- Hollad Siee Pbliher Aerda 6. [7] V. Lahiaha A. S. Vaala heor o raioal Diereial Eaio ad Appliaio Co.Appl.Aal [8] V. Lahiaha A. S. Vaala Bai heor o raioal Diereial Eaio ad Appliaio Nol. Aal [9] V. Lahiaha A. S. Vaala Geeral Uiee ad oooe eraie eh ie or raioal Diereial Eaio Appl. ah. Le [] V. Lahiaha S. Leela ad J. V. Dei heor o raioal Dai Se Cabridge Sieii Pbliher Cabridge U.K 9. [] J. A. Naware oooe ehod i raioal Diereial Eaio ad Applia io dooral hei Dr.Babaaheb Abedar arahwada Uieri. [] J.A.Naware Exiee Rel or Capo raioal Bodar Vale Proble Uig ixed Poi heore er.j. ah.ah. Si [] J. A. Naware Exiee ad Uiee Rel or raioal Diereial EaioVia oooe ehod Bll. arahwada ah. So [] J. A. Naware D. B. Dhaigde Exiee ad Uiee o olio o Riea- Lioille raioal Diereial Eaio wih egral Bodar Codiio. Jor. Noliear Siee. -5. [5] J. A. Naware D. B. Dhaigde oooe eraie Shee or Se o Riea- Lioille raioal Diereial Eaio wih egral Bodar Codiio ah. odellig Sie. Cop. Spriger-Verlag eraioal orgaizaio o Sieii Reearh 6 P a g e where. he b ieali 7 here exi h ha =. Le Y C J R :. he operaor : Y C J R i oio ad opleel oio. B he hoie o Y here exi o Y h ha = λ or oe λ. A a oeee o he oliear aleraie o Lera-Shader pe [5] we dede ha ha a ixed poi i Ȳ whih i he olio o he BVP -.!

7 Paper preparaio gidelie or OSR Joral o Egieerig [6] J. A. Naware D. B. Dhaigde oooe ehie or iie Se o Capo raioal Diereial Eaio wih Periodi Bodar Codiio D. Coi. Di. pl. S. A a - 5. [7] J.A.Naware D.B.Dhaigde Bodar Vale Proble or Diereial Eaio o No-ieger Order olig Capo raioal Deriaie Ad. S. Coe. ah [8] J.A.Naware D.B.Dhaigde Exiee ad Uiee o Solio o Diereial Eaio o raioal Order wih egral Bodar Codiio J. Noliear. Si. Appl [9] J.A.Naware N.B.Jadha D.B.Dhaigde oooe eraie ehie or iie Se o Riea-Lioille raioal Diereial Eaio wih egral Bodar Codiio eraioal Coeree o aheaial Siee Eleier 5-8. []. Podlb raioal Diereial Eaio Aadei Pre Sa Diego 999. [] J. D. Rairez A. S. Vaala oooe eraie ehie or raioal Diereial Eaio wih Periodi Bodar Codiio Op. ahe []. Wag. Xie Exiee ad Uiee o raioal Diereial Eaio wih egral Bodar Codiio he J. Nol. Si. Appl [] S. Zhag oooe eraie ehod or iial Vale Proble olig Riea- Lioille raioal Deriaie Nol. Aal eraioal orgaizaio o Sieii Reearh 7 P a g e